Subgroups — C₁₃×#9851 #15696190

Order 3328.

492 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1962	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₂²	1	1	yes	characteristic
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₁₃×#498	C₄	1	1	yes	characteristic, complemented
11	C₁₃×#492	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#487	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#486	C₄	1	1	yes	characteristic, complemented
14	C₁₃×#465	—	2	2	no	—
15	C₄×C₁₃×SD₁₆	—	2	2	no	—
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#423	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#104	—	2	2	no	—
19	C₂×C₁₃×#97	—	2	2	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₄³×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#339	—	2	2	no	—
24	C₁₃×#335	—	2	2	no	—
25	C₁₃×Q₈⋊C₈	—	2	2	no	—
26	C₁₃×D₄⋊C₈	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂×C₁₃×SD₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#125	—	2	1	no	—
38	C₁₃×#124	—	2	1	no	—
39	C₁₃×C₂²⋊Q₈	—	2	1	no	—
40	C₁₃×C₂²⋊Q₈	—	2	1	no	—
41	C₁₃×#121	—	2	1	no	—
42	C₁₃×#121	—	2	1	no	—
43	C₄×Q₈×C₁₃	—	2	1	no	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	—	2	1	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₂×C₄²×C₁₃	—	2	1	no	—
49	C₂×C₄²×C₁₃	—	2	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#106	—	2	2	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×#104	—	4	4	no	—
58	C₁₃×#103	—	2	2	no	—
59	C₁₃×#102	—	2	2	no	—
60	C₁₃×#97	—	2	8	no	—
61	C₄×C₈×C₁₃	—	2	1	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	D₄	1	2	yes	—
66	#9851	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	4	2	no	—
68	C₁₃×#41	—	4	2	no	—
69	C₂×Q₈×C₁₃	—	2	1	no	—
70	C₂×Q₈×C₁₃	—	2	1	no	—
71	C₂×Q₈×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₁₃×SD₁₆	—	4	4	no	—
82	C₁₃×M₁₆	—	4	2	no	—
83	C₁₃×M₁₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	1	no	—
92	C₁₃×C₄⋊C₄	—	2	1	no	—
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	2	4	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	#41	1	1	yes	characteristic
111	#3167	C₂×C₁₃	1	1	yes	characteristic
112	#1994	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1962	C₂×C₁₃	1	1	yes	characteristic
114	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1493	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#1492	C₂×C₁₃	1	1	yes	characteristic
118	C₂³×C₁₃	—	4	1	no	—
119	C₂³×C₁₃	#121	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	8	1	no	—
136	C₂×C₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₁₃	#137	1	2	yes	—
145	C₈×C₁₃	—	2	4	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	8	1	no	—
148	C₈×C₁₃	—	8	1	no	—
149	C₈×C₁₃	#104	1	4	yes	complemented
150	#533	C₂²×C₁₃	1	1	yes	characteristic
151	#532	C₂²×C₁₃	1	1	yes	characteristic
152	#498	C₄×C₁₃	1	1	yes	characteristic, complemented
153	#492	C₄×C₁₃	1	1	yes	characteristic, complemented
154	#487	C₄×C₁₃	1	1	yes	characteristic, complemented
155	#486	C₄×C₁₃	1	1	yes	characteristic, complemented
156	#465	—	2	2	no	—
157	C₄×SD₁₆	—	2	2	no	—
158	#423	C₂²×C₁₃	1	1	yes	characteristic
159	#423	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#104	—	2	2	no	—
161	C₂×#97	—	2	2	no	—
162	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₄³	C₂²×C₁₃	1	1	yes	characteristic
165	#339	—	2	2	no	—
166	#335	—	2	2	no	—
167	Q₈⋊C₈	—	2	2	no	—
168	D₄⋊C₈	—	2	2	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	8	1	no	—
172	C₂²×C₁₃	—	8	1	no	—
173	C₂²×C₁₃	—	8	1	no	—
174	C₂²×C₁₃	#390	1	1	yes	characteristic
175	C₂²×C₁₃	#468	1	2	yes	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	8	1	no	—
180	C₄×C₁₃	—	8	1	no	—
181	C₄×C₁₃	—	8	1	no	—
182	C₄×C₁₃	C₂×#104	1	2	yes	—
183	C₄×C₁₃	#440	1	2	yes	—
184	C₄×C₁₃	#444	1	2	yes	—
185	C₄×C₁₃	#496	1	1	yes	characteristic
186	C₄×C₁₃	#497	1	1	yes	characteristic
187	C₂×#41	—	2	1	no	—
188	C₂×SD₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	D₄×C₁₃	1	2	yes	—
193	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
196	#127	C₂×C₄×C₁₃	1	1	yes	characteristic
197	#125	—	2	1	no	—
198	#124	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	C₂²⋊Q₈	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	#117	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	D₄×C₁₃	1	2	yes	—
211	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
212	#106	—	2	2	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	#104	—	4	4	no	—
218	#103	—	2	2	no	—
219	#102	—	2	2	no	—
220	#97	—	2	8	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
224	C₄×C₈	D₄×C₁₃	1	2	yes	—
225	C₄×C₈	D₄×C₁₃	1	2	yes	—
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	16	1	no	—
228	C₂×C₁₃	#1962	1	1	yes	characteristic
229	C₂×C₁₃	#1973	1	1	yes	characteristic
230	C₂×C₁₃	#1975	1	1	yes	characteristic
231	#41	—	4	2	no	—
232	#41	—	4	2	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
245	SD₁₆	—	4	4	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
254	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₁₃×#41	1	1	yes	characteristic
275	C₁₃	#9851	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	4	1	no	—
277	C₂³	C₁₃×#121	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
301	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
302	C₂×C₄	C₁₃×#137	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	8	1	no	—
306	C₈	—	8	1	no	—
307	C₈	C₁₃×#104	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₁₃×#390	1	1	yes	characteristic
314	C₂²	C₁₃×#468	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	C₂×C₁₃×#104	1	2	yes	—
322	C₄	C₁₃×#440	1	2	yes	—
323	C₄	C₁₃×#444	1	2	yes	—
324	C₄	C₁₃×#496	1	1	yes	characteristic
325	C₄	C₁₃×#497	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	C₁₃×#1962	1	1	yes	characteristic
329	C₂	C₁₃×#1973	1	1	yes	characteristic
330	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₁₃×#1994	C₂	semidirect
4. C₄×C₁₃×#97	C₂	semidirect
5. C₁₃×#1567	C₂	semidirect
6. C₁₃×#1493	C₂	semidirect
10. C₁₃×#498	C₄	semidirect
11. C₁₃×#492	C₄	semidirect
12. C₁₃×#487	C₄	semidirect
13. C₁₃×#486	C₄	semidirect
66. #9851	C₁₃	direct (N × H)
112. #1994	C₂×C₁₃	semidirect
114. C₄×#97	C₂×C₁₃	semidirect
115. #1567	C₂×C₁₃	semidirect
116. #1493	C₂×C₁₃	semidirect
149. C₈×C₁₃	#104	semidirect
152. #498	C₄×C₁₃	semidirect
153. #492	C₄×C₁₃	semidirect
154. #487	C₄×C₁₃	semidirect
155. #486	C₄×C₁₃	semidirect
275. C₁₃	#9851	direct (N × H)
307. C₈	C₁₃×#104	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3167	2
C₁₃×#1994	2
C₁₃×#1962	2
C₄×C₁₃×#97	2
C₁₃×#1567	2
C₁₃×#1493	2
C₁₃×#1492	2
#9851	13

Subgroups — C13×#9851 #15696190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9851 #15696190